To the best of our knowledge, only a few articles investigate convergence rates. In [1]} a truncation approach was adopted; this can be formulated as a two-step procedure. The authors first fix \(({\phi }_\lambda )_{\lambda \in \Lambda }\)  an orthonormal basis of the space considered, then they choose \(M\)  components, i.e. \(\lbrace \phi _{\lambda _1},\dots ,\phi _{\lambda _M}\rbrace \) , and they project the curves on those \(M\)  components. Finally they compute the eigenfunction after the projection step. The obtained rates using this approach are at best of the order of \(O(\frac{M^3}{n})\) .
In ([2]} and [3]}) a nonparametric approach was adopted, based on kernel estimation, assuming that the covariance is twice differentiable. This results in rates depending on the size of the bandwidth \(h\)  of the order of \(O_p(h^4 +\frac{\log (n)}{nh})\) .
In [4]} we showed that in the univariate case, assuming \(Z\)  to be a \(\alpha -\) Hölder continous function, the non-penalised estimator converges at minimax rates (\(O(n^{-1})+O(p^{-2\alpha })\) ), however in the \(D\) -dimensional context, the same approach gives rates of order (\(D^2 n^{-1}+D^2p^{-2\alpha }\) ). So, in a high dimensional context where \(D\gg n\) , those rates are non-relevant.
